Non-Attractive Kernels of Singularity

• Non-attractive kernels of singularity are defined by weak or integrable singularities that lack dominant attractive behavior and standard scaling properties.
• They are utilized in nonlocal PDEs, harmonic analysis on homogeneous spaces, and mean-field particle systems to establish novel regularity and convergence results.
• Their analysis necessitates innovative tools, such as intrinsic Orlicz–Besov spaces and dyadic decompositions, to overcome challenges in classical singular integral techniques.

Non-attractive kernels of singularity are a class of singular kernels distinguished by the absence of homogeneity, strong repulsive or non-sign-definite interaction, or reduced scaling and cancellation features. This concept spans several domains: mean-field limits for singular particle systems, nonlocal operators in PDE, harmonic analysis on homogeneous spaces, singular integral characterization of geometric properties, and operator theory in analytic function spaces. The term "non-attractive" is often used to indicate that the kernel lacks dominant attractive behavior or standard scaling, and may be weakly singular, even, positive, or non-antisymmetric.

1. General Definitions and Structural Characteristics

Non-attractive kernels of singularity are functions with singularities that do not induce strong attraction or scaling symmetry near the singularity. Typical examples are:

• Weakly singular jump kernels in nonlocal operators, such as those for subordinate Brownian motions, which do not admit Lévy–stable scaling (Hu et al., 2023).
• Even, nonnegative kernels (as opposed to antisymmetric, attractive Riesz-type kernels) in geometric measure theory, notably on non-Euclidean groups like the Heisenberg group (Chousionis et al., 2016).
• Interaction kernels for mean-field particle systems, $K(x) = |x|^{-\alpha}$ with $0 < \alpha < d-1$, satisfying non-attraction conditions: $(K(x)-K(-x))\cdot x \geq 0$ (Höfer et al., 17 Sep 2025).
• Integrable singular kernels in harmonic analysis on compact manifolds, e.g., the Riesz kernel $d(x, y)^{-s}$ for $x=y$, with $d$ a metric lacking standard extension/smoothness at the diagonal (Bilyk et al., 29 Oct 2024).

These kernels are characterized by their weak or integrable singularities, absence of cancellation or scaling required for classical singular integral techniques, and non-attractive or non-antisymmetric structure.

2. Nonlocal Operators with Low Singularity Kernels

Operators of the form

$$\mathcal{L} u(x) = \int_{\mathbb{R}^d} [u(x+z) - u(x)] a(x,z) J(z)\,dz$$

enable analysis of jump-type Lévy processes with kernels $J(z)$ exhibiting low-order singularity near $z=0$ (Hu et al., 2023). Here,

• The kernel $J(z)$ may fail to scale as $|z|^{-d-\alpha}$ (for $\alpha\in(0,2)$), and instead models slowly varying or "logarithmic Laplacian" behavior.
• Standard harmonic analysis methods (e.g., Littlewood–Paley decomposition) are ineffective.
• The absence of standard scaling motivates the definition of generalized Orlicz–Besov spaces using intrinsic dyadic decompositions calibrated to the underlying Lévy process (via increasing functions $v(r)$ linked to the Lévy exponent).

For such operators:

• Regularity theory (e.g., Schauder estimates) uses intrinsic metric structures and Morrey-type inequalities in Orlicz–Besov norms.
• The martingale problem is well-posed under minimal conditions on coefficients and the jump kernel, allowing construction of strong Markov processes without attractive jump mechanisms.
• Krylov-type estimates for occupation times are achieved, crucial for probabilistic uniqueness and regularity.

3. Singular Kernels in Harmonic Analysis and Geometry

In harmonic analysis on two-point homogeneous spaces, considering kernels with singularity at the diagonal, e.g. Riesz kernels $K(x, y) = d(x, y)^{-s}$, necessitates a redefinition of positive definiteness (Bilyk et al., 29 Oct 2024). Key structural aspects:

• The positive definiteness is determined by nonnegativity of Fourier–Jacobi expansion coefficients.
• Comparative analysis of the geodesic versus chordal metrics reveals that, for the geodesic metric on projective spaces, Riesz kernels may lose positive definiteness at fixed $s>0$ due to stronger singularity and absence of Euclidean embedding. There exists a "critical exponent" $s_d(\mathbb{F})$ depending on the dimension and underlying field, above which positive definiteness fails.
• Energy minimization and Schur product results extend to singular kernels via integral criteria, not matrix positivity.

These findings illustrate that positive definiteness for singular kernels is intimately connected with the nature of the singularity and the geometric embedding properties.

4. Non-Attractive Singular Kernels in Particle Systems: Mean-Field and Chaos

For mean-field systems with binary interaction kernel $K(x) = |x|^{-\alpha}$ and $\alpha < d-1$, the propagation of chaos and quantitative mean-field limit depend crucially on whether the kernel is attractive or not (Höfer et al., 17 Sep 2025). The "non-attractive" property $(K(x)-K(-x))\cdot x \geq 0$ yields:

• It is sufficient to control the distance to the next-to-nearest neighbor particle, rather than the global minimal particle separation. This relaxation enables proof of mean-field behavior under much weaker initial conditions.
• Improved parameter range for propagation of chaos: for i.i.d. initial data with bounded density, propagation holds for $\alpha < \min\{(d-1)/2, (2d-3)/3\}$, exceeding previous thresholds by Hauray and others.
• Quantitative convergence in Wasserstein distance and lower bounds on particle separation are deduced with explicit control over singular sums: $$S_{\alpha+1,\delta} = \sup_i \sum_{j\neq i, |X_i-X_j|>\delta} \frac{1}{|X_i-X_j|^{\alpha+1}}$$
• The non-attraction condition greatly simplifies the bootstrap analysis required for deterministic and probabilistic estimates.

This framework highlights how sign structure and singularity of the kernel influence collective dynamics and statistical independence in large systems.

5. Rectifiability and Singular Integrals without Cancellation

In the Heisenberg group, even, nonnegative, and $-1$–homogeneous kernels, such as $K_1(p) = [\Omega(p)]^8/N(p)$ and $K_2(p) = [\Omega(p)]^2/N(p)$ with vertical weight $\Omega$ and Korányi norm $N$, serve as geometric probes for $1$–rectifiability (Chousionis et al., 2016). Notable features:

• Unlike the Euclidean theory where antisymmetric kernels are required for $L^2$–boundedness, in the Heisenberg setting, positivity and evenness suffice.
• $L^2$–boundedness with the stronger kernel $K_1$ is necessary for rectifiable curves; boundedness with $K_2$ is sufficient and characterizes $1$–rectifiability.
• This result constitutes the first non-Euclidean instance where non-antisymmetric singular kernels can detect geometric regularity directly via singular integral estimates.

The absence of cancellation in the kernel necessitates new strategies exploiting the group geometry, particularly the neutralization of vertical component along horizontal curves.

6. Analytical and Probabilistic Implications

Non-attractive, weakly singular kernels require development of novel analytic and probabilistic tools:

• Morrey-type inequalities and Orlicz–Besov spaces cater to regularity analysis where classical $L^p$ or Besov spaces are unsuitable (Hu et al., 2023).
• The solution theory for elliptic and parabolic PDEs extends to these contexts, with martingale solutions well-posed under mild conditions, and occupation times controlled by generalized norm estimates.
• On compact symmetric spaces, energy minimization for such kernels is classified by spectral positivity, fundamentally altered by the singularity order and underlying metric (Bilyk et al., 29 Oct 2024).
• In many-particle systems, chaos propagation is established with sharper thresholds, demonstrating the critical impact of "non-attractive" singularity structure on scaling limits and statistical independence.

7. Future Research Directions

Open problems and future research involve:

• Determination of sharp critical exponents for positive definiteness of singular Riesz kernels on diverse homogeneous spaces and in high dimensions.
• Extending intrinsic Orlicz–Besov decompositions to nonlocal operators with even weaker (or variable) singularities and to non-Euclidean settings.
• Refinement of singular integral characterizations for rectifiability and removability in general Carnot groups or higher codimension settings, optimizing vertical weight powers.
• Exploration of the impact of non-attractive kernel structure in stochastic systems, especially for degenerate or non-Markovian jump processes.

Continued investigation of non-attractive kernels of singularity is essential for progress in singular operator theory, geometric measure classification, stochastic analysis, and statistical physics.